You need to buy an American Strat or Tele, so tell your wife you ABSOLUTELY have to have one.
There. I told you.
The better and more sensitive and sustaining the guitar you get- the better said Strat or Tele are- the more it will be capable of sympathetic string vibration. So, that won't make that go away, it will only make you practice more.

Seriously, damping and muting out unwanted sympathetic string vibration is all down to your hands and fingers. (And a guitar's tendency to sympathetic string vibration usually indicates a good, healthy guitar.) I tend to let the side and heel of my picking-hand's palm rest on the strings, somewhat like when "palm muting" for chunk rhythms and Al Dimeola approved "Elegant Gypsy"-esque muted picking, and any unused fingers or thumbs on either hand go on and/or in-between the strings where they'll conveniently reach.
This takes some thought and focus at first, but the longer you're at it, the more it becomes automatic second-nature, to the point of having to take a moment to figure out how you've been doing it all along if, for instance, you want to explain how it's done to someone.

This has been touched upon in some of the online lesson-videos that Arlen Roth has been doing for the Gibson Lifestyle 'site and e-newsletters, like "
Slide Guitar Blocking and Damping Techniques".